M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

Quantification of ATF Concept Assessment Factors Using Multiphysics Simulations

James B. Tompkins,∗ Ryan G. McClarren,∗ Jason D. Hales †

∗Department of Nuclear Engineering, Texas A&M University, 3133 TAMU, College Station, TX 77843†Fuels Modeling & Simulation, Idaho National Laboratory, 2525 N. Fremont Ave., Idaho Falls, ID 83415

tompkins@tamu.edu, rgm@tamu.edu, jason.hales@inl.gov

Abstract - In persuit of a quantitative method of comparing accident tolerant fuel concepts to traditional fuelbehaviors, a method of applying failure probabilities in an existing fuel assessment framework is proposed. Todemonstrate this process, an example case is undertaken by utilizing coupled multiphysics codes to investigateuranium silicide and uranium oxide fuel behaviors during a postulated reactivity insertion accident. Anupper limit on each fuel system’s surface cladding temperature is used to classify pass and fail cases onperturbed simulations. Sensitivity analysis reduces the dimensionality of the problem and a global geometricclassification method predicts the limit surface separating pass and fail cases in the hyperspace made upof sensitive input variables. Numerically integrating this surface yields the probability of failure of the fuelconcepts. Results demonstrate the method’s viability as a fuel evaluation technique, but ongoing work willallow for quantification of proposed concept assessment factors.

I. INTRODUCTION

In the wake of events at Fukushima Daiichi in 2011, therehas been renewed worldwide interest in the development of nu-clear fuel with greater tolerance for safe operation in accidentconditions. Recent collaborative efforts between laboratories,industry, and universities to develop accident tolerant fuels(ATF) have produced numerous concepts, and an aggressiveATF implementation schedule has yielded several candidatematerials with promising experimental results. Physical in-vestigation alone, though, cannot provide the extensive datanecessary for a thorough assessment. To aid in the analysisof ATF concepts, the Nuclear Energy Advanced Modeling &Simulation (NEAMS) program has directed a High ImpactProblem aimed at supporting the Advanced Fuels Campaign byutilizing multiscale, multiphysics fuel performance modelingand simulations capabilities [1]. By employing state-of-the-artcomputational tools in ATF appraisals, more complete insightsinto fuel concepts’ behaviors are gained.

Results from advanced analyses of ATF concepts are notdirectly comparable with traditional fuels’ performance met-rics as the new materials are designed to react to system tran-sients differently. A method of accurately comparing fuelbehaviors and drawing conclusions about the advantages anddisadvantages of each fuel system should utilize results fromboth fuels under similar reactor system conditions while ac-comodating differences in fuel designs. Using a developedevaluation framework, we propose a methodology to calculateConcept Assessment Factors (CAFs), providing a numericscore for ATF candidate reviewers to consider in their ap-praisals. Simulations of silicide and oxide fuel elements dur-ing a reactivity insertion accident (RIA) serve as an exampleto demonstrate the methodology. Models from BISON cou-pled to RELAP-7 provide thermomechanical and two-phasethermohydraulic results for the posed problem, and modelinput variables are perturbed to investigate sensitivity of pa-rameters as well as determine a hypersurface separating passand fail cases. The failure probability of each fuel system canthen be calculated and a comparison will yield a quantitatively

determined Concept Assessment Factor.

II. THEORY AND CONTRIBUTING WORK

The following framework employed to assess ATF con-cepts already details methods an evaluator might use to de-termine the strengths of various fuel systems, but the basemetrics are determined subjectively. In order to reduce theframework’s reliance upon qualitatively determined metrics,the addition of factors arrived at through multiphysics sim-ulation results is presented. An overview of the technicalperformance evaluation framework as well as a description ofthe demonstration problem to illustrate CAF quantification arespecifed in this section. The tools utilized to solve the demon-stration problem are also outlined with a brief accounting ofthe methods utilized in these codes and physical quantitiesable to be modeled. Lastly, one may find a summary of statisti-cal and machine learning methods employed in quantificationof CAFs in this section.

1. Concept Assessment Framework

To quantitatively determine the viability of ATF concepts,a method of evaluating advanced fuel candidates in a varietyof operational conditions was developed by Bragg-Sitton et al.[2]. The methodology outlines methods of evaluation in threephases of technological development: Feasibility Assessment(proof-of-concept), Development and Qualification (proof-of-principle), and Commercialization (proof-of-performance)[2]. At any stage of investigation a Candidate Fuel SystemAttributes Assessment Table is used to evaluate fuel systemconcepts [2]. This table divides attributes into PerformanceRegimes including fabrication, normal operation, design basisaccidents, severe accidents, and fuel storage with associatedregime ranks and weights in ascending order of perceived im-portance [2]. Each regime is further divided into PerformanceAttributes representing behaviors and interactions necessaryto consider in that Performance Regime with each attributehaving an associated rank and weight [2]. The attributes are in-

M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

dividually compared between the concept and traditional UO2-zircaloy fuel system to obtain Concept Assessment Metrics(CAMs) that contribute to benefits and vulnerabilities sepa-rately [2]. Assessment scores for candidate fuel systems aredetermined for benefits and vulnerabilities by summing theweighted CAMs:

ScoreBenefit =

V∑i=I

wi

h∑j=a

w jCAMBenefit (1a)

and

ScoreVulnerability =

V∑i=I

wi

h∑j=a

w jCAMVulnerability , (1b)

where wi is the fractional regime weighting and w j is theattribute weight within the regime [2]. The weighting factorslisted in the developed table were determined with input fromacademic and industry professionals [2]. While the workmakes a point of determining the CAMs subjectively, wepropose using advanced simulation capabilities to quantifyConcept Assessment Factors (CAFs) that fit into the technicalperformance evaluation framework in place of the subjectivemetrics [2].

2. Concept Assessment Factor Quantification

The projected workflow for quantifying Concept Assess-ment Factors is shown in Figure 1. Diamond shaped nodes inthis figure denote decisions to be made at the beginning of theCAF quantification process. Each decision limits further eval-uation choices until some physical parameter(s) contributingto quantification of a performance attribute and a failure con-dition for the parameter are determined. After first selectinga candidate fuel system to investigate, one then chooses theregime in which the system operates in the simulation. A per-formance attribute aiding in evaluating the operation regimeis next selected from attribute assessment table Identifiers.The multiphysics capabilities of the tools selected determineboth the operation regimes and performance attributes ableto be modeled, thus the tools selected largely determine theCAFs that can be computed. Physical properties prominentlyinfluencing the chosen performance attribute are selected fromquantities the computational tools are able to model, and a fail-ure criterion for physical properties can be determined fromeither an actual limit (e.g. boiling point for fluid temperature)or safety factor on a quantity limit. In addition to nominalmaterial properties required to model the variable of interestin the operation regime chosen, distributions for these inputvariables in the computational tools are necessary to performthe sensitivity and classification analyses to quantify CAFs.

A demonstration case is undertaken to demonstrate thisprocess. Employing coupled BISON and RELAP-7 simu-lations to solve fuel element thermomechanics and coolantchannel thermal hydraulics allows for a wide range of CAFsto be quantified as the coupled codes are able to model myriadphysical parameters affecting performance attributes of inter-est in several operation regimes. In addition to traditional fuelconcepts, BISON has already implemented thermomechanical,

material, and behavioral models for ATF concepts such asFeCrAl alloy cladding and silicide fuels [3]. Subsection II. 3.Computational Tools contains descriptions and overviews ofthe capabilities for these tools; the remainder of this sectionis devoted to characterizing the demonstration case chosen toquantify a CAF.

A case must be chosen for a concept with methods alreadyimplemented in operating regimes of interest in the selectedtools. An assessment of the uranium silicide (U3Si2) fuelconcept during a reactivity insertion accident is both a novelproblem which coupled BISON and RELAP-7 can solve andserves to evaluate a CAF for the thermal behavior of U3Si2in the postulated accident regime. Numerous quantities ofinterest are available to compare oxide and silicide thermal be-havior performance, but to limit the focus of the demonstrationcase, the maximum outer cladding temperature is chosen. Afailure condition of 1140 K on the outer surface of the claddingallows for a 0.45 safety factor on the melting temperature ofzircaloy and corresponds to the middle of the elastic modulus’interpolation temperature evaluation range [4].

Simulating a single 4.435 meter PWR reactor fuel ele-ment and coolant channel, the system starts up to a low power,steady state condition after which the RIA occurs and thesystem recovers. The power history during the RIA is de-rived from scaled experimental control rod ejection data in aBrookhaven National Laboratory report (see Figure 2) [5]. Thepower profiles with which each fuel is simulated are shown inFigure 2. The differing power profiles are necessary to obtainthe same nominal maximum outer cladding temperature ofapproximately 1130 K, but initial results are also comparedwith UO2 results with the same power profile of the silicidefuel.

To perform sensitivity analysis and perturb input variablesto create a limit surface, distributions for the variables needto be specified and sampled. Parameter distributions from thePhase-II OECD NEA RIA fuel codes benchmark are employedin this analysis (Table I) [6]. Variables include pressures, ve-locities, temperatures, densities, physical measurements, andscaling factors for thermal parameters ( fk,fuel, fk,clad, fcp,fuel).

Fig. 1. Proposed workflow to quantify CAFs. Selection ofdemonstration problem characteristics are in parentheses.

M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

Fig. 2. Linear heat rate provided as a power history to BISONduring the RIA.

Parameter Unit Mean Std. Dev.

pliq.,init. Pa 15.5 × 106 0.075 × 106

vliq.,init. m/s 4.00 0.04Tliq.,init K 280 1.5ρfuel kg/m3 10431.0 51.5dclad_out m 9.4 × 10−3 0.01 × 10−3

dclad_in m 8.26 × 10−3 0.01 × 10−3

pplen.,init. Pa 2.0 × 106 0.05 × 106

Rclad m 0.5 × 10−6 0.25 × 10−6

Rfuel m 2.0 × 10−6 1.0 × 10−6

fk,fuel - 1.0 0.05fcp,fuel - 1.0 0.05fk,clad - 1.0 0.015

TABLE I. Distributions for parameters to perturb for RIA [6].

3. Computational Tools

Figure 1 demonstrates the necessity of selecting appro-priate computational tools. Codes utilized to simulate fuelbehaviors need to be able to accurately simulate fuel conceptresponses in various operation regimes and calculate parame-ters affecting chosen performance attributes of interest whilebeing robust enough to simulate failure conditions. CoupledBISON and RELAP-7 calculations meet all of the above re-quirements and perturbed models from RAVEN give insightsinto simulation uncertainties. With RAVEN driven BISONand RELAP-7 simulations, Concept Assessment Factors areable to be quantified.

One of NEAMS’ tools of choice for fuel performancemodeling is Idaho National Laboratory’s (INL) BISON codewhich integrates developed nuclear material and behavioralmodels in the heat transfer and solid mechanics modules avail-able in the MOOSE finite element framework on fuel elementmeshes to effectively model fuel system thermomechanics [3].In addition to traditional fuel concepts, BISON has already im-plemented thermomechanical, material, and behavioral models

for ATF concepts such as FeCrAl alloy cladding and silicidefuels [3]. The developed BISON models for both fuel casescontain methods for fuel and cladding solid mechanics, heattransfer , thermal and mechanical contact, and plasticity cladmodels. As BISON has been developed using the MOOSEframework, it can be easily coupled with other codes devel-oped in the framework provided the physics they model areable to be employed as a boundary condition or source inBISON models to extend its multiphysics capabilities [7].

RELAP-7, an INL development code for thermal hy-draulics built on the MOOSE framework, can be coupledto BISON through the MAMMOTH application to calculatecoolant channel conditions during simulated reactor operationsand pass temperature and pressure to the models in BISON as aboundary condition [7] [8]. RELAP-7 simulations use a sevenequation, two-phase fluid flow model with various stabilizationtechniques and have integrated probabilistic risk assessmentcapabilties [8]. Our coupled simulation uses control logic tovary coolant inlet velocity and density as the system starts up,and the entropy viscosity method is employed for stabilization.While BISON does have a coolant channel model capableof calculating heat transfer from the outer clad wall into thecoolant, it is only able to simulate thermal energy depositionin the coolant during steady state and slow operating transientconditions facilitating the choice of the coupled simulation fora RIA [3]. By coupling to RELAP-7, the thermal hydraulicsimulation response capabilities are greatly expanded in BI-SON models, especially in fast transient cases such as a RIA.

Coupling these codes enables modeling of fuel elementsand coolant channel response during operational transientsusing thermomechanical and thermal hydraulics methods tosimulate a fuel system’s behavior, but neither on their own al-low a user to perturb input variables, necessary when exploringoutput response to predictors. RAVEN allows users to providedistributions for input variables from which it can then samplefor use in generated input files for perturbed code runs [9]. Itscapabilities include scheduling multiple jobs to run in paralleland constructing statistical models from collected output for avariety of analyses [9]. Developed statistical modeling capa-bilities are supplemented by the scikit-learn Python machinelearning module which is employed in this work for its linearregression models used to perform sensitivity analysis [9]. Inaddition, RAVEN’s adaptive sampler is used to determine thelimit surface separating pass and failure states by utilizingsupport vector machines (SVM), a geometric classificationmethod [9].

4. Statistical&Machine Learning Methods

The linear model fit with the least squares method couldbe called the most ubiquitous paradigm in the zeitgeist ofcontemporary statistics. Making large assumptions about thestructure of predictive solutions, the linear model yields stablesolutions applicable to a a great number of datasets [10]. Theoutput Y of a model is predicted from a provided vector ofinputs XT =

(X1, X2, . . . , Xp

)via

Y = β0 +

p∑j=1

X j β j , (2)

M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

where the term β0 is the intercept [10]. In order to fit the modelto training data, the coefficients β are chosen to minimize theresidual sum of squares

β = argminβ

N∑

i=1

(yi − β0 −

p∑j=1

xi jβ j)2

[10]. (3)

The residual sum of squares’ minimum always exists (as it is aquadratic function), but this minimum may not be unique [10].By imposing penalties on the size of least squares coefficientestimates, shrinkage methods arise. Ridge regression appliesa sum-of-squares (L2) penalty to the coefficients

βridge = argminβ

N∑

i=1

(yi − β0 −

p∑j=1

xi jβ j)2 + λ

p∑j=1

β2j

(4)

which shrinks β values both towards zero and other coefficients,alleviating the high variance of linear regression models withmany correlated variables [10]. The complexity parameterλ ≥ 0 determines the amount of shrinkage applied to thecoefficients [10]. Next, the lasso’s coefficient estimation isgiven by

βlasso = argminβ

12

N∑i=1

(yi − β0 −

p∑j=1

xi jβ j)2 + λ

p∑j=1

|β j|

(5)

where again, the λ ≥ 0 is the complexity parameter used tovary shrinkage [10]. Notice that the L2 penalty

∑p1 β

2j from

ridge regresssion is now an L1 sum of absolute values penalty∑p1 |β j| for the lasso [10]. The absolute value contraint causes

some coefficients to be exactly zero for large λ’s living upto its name by "lassoing" sensitive parameters, thus selectinga subset of input variables [10]. The elastic net penalty is acompromise between the lasso penalty’s indifference to thechoice among a set of sensitive but correlated variables and theridge penalty’s tendency to shrink the coefficients of correlatedvariables toward one another

p∑j=1

(α|β j| + (1 − α)β2

j

)(6)

where α is a parameter that determines the weights of thepenalties [10].

One challenge in utilizing shrinkage methods to buildpredictive models is the selection of complexity and weightingparameters. The values of these parameters yielding the bestfit can be determined via cross-validation, leaving a portionof the data out when training models then using the left outdata to test the accuracy of the resultant model by comparingprediction values with the quantity of interest and iterativelydetermining the penalties that reduce model error by the great-est amount [10].

How then is the training data obtained? Looking backto Figure 1, the perturbed parameters chosen as input for thenominal models feed into the sensitivity analysis. However,generating models with the perturbed parameters is a prob-lem of scale: each data point obtained is computationally

expensive, but model error is reduced by exploring a greateramount of the sample space. To sample a variety of inputvalues while being conservative with computational resources,the sampling scheme utilized is the latin hypercube, a gridbased approach in which each sample is individually limitedto one axis-aligned hyperplane [11]. A major advantage ofthis method is the independence of the number of samplesupon the number of inputs to explore [11].

After training the models, the sensitivity of chosen out-put parameters to input variables can be determined from themodel coefficients. The coefficients produced weight how im-portant each associated input variable is in predicting output,but the coefficients must be converted to the same units beforebeing compared. By multiplying coefficients by their asso-ciated variables’ sample distribution parameters, the unitlesscoefficients rank, in descending magnitude, input variableshaving the greatest effect upon the model’s output.

With sensitive parameters identified and a failure condi-tion available, support vector machines (SVM) can be used tofind a limit surface for the classification problem. The goal isto determine the hypersurface in a subset of sensitive parame-ters separating pass and fail cases determined by the parameterof interest’s failure condition. The most important parametersidentified in the sensitivity analysis can be used as a subsetselection of perturbed input variables to minimize the amountof sample data required to accurately identify input behav-iors affecting the quantity of interest. To find a limit surface,support vector classifiers, a global geometric classificationmethod, find maximum margin hyperplanes by constructinglines from a set of points [10]. These lines are then used todetermine a limit surface via an optimization scheme

minβ,β0||β|| (7)

subject to yi(xTi β + β0) ≥ 1, i = 1, . . . ,N [10].

This method can be extended to use nonlinear functions such aspolynomials or splines for limit surfaces through kernelizationby replacing the dot product between two vectors with a moregeneral function [10]. RAVEN can use SVM to construct aninitial limit surface, then sample around the surface to refine it[9].

Once a limit surface that separates the input space intosafe and unsafe hypergeometries is determined, the unsaferegion can be numerically integrated to quantify the probabilityof failure for a particular accident in the fuel types investigated.The difference between the ATF and oxide failure probabilitiesdirectly correlates to the disparity in their unsafe hypervolumeswhich will yield a benefit CAF if the ATF concept’s probabilityis smaller and a vulnerability if it is larger.

III. RESULTS AND ANALYSIS

1. Nominal Results

Figure 3 presents the axial clad surface temperature at thetime of the largest maximum temperature for the silicide andtwo oxide cases. The oxide case results shown are run at thetwo different power profiles from Figure 2. The silicide hasa higher maximum clad temperature of 1130 K at an earlier

M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

Fig. 3. Axial fuel concept temperature distributions.

Fig. 4. Axial ratio of fuel centerline temperature to meltingtemperature.

Fig. 5. Volume fraction of vapor at coolant channel outlet overtime.

time compared to the oxide fuel which has a maximum surfacecladding temperature of 998 K. This result makes sense asthe advertised advantage of silicide is the higher heat transfercoefficient, but it does call into question whether an accuratecomparison is being drawn between the two fuel systems. Theassumption that U3Si2 and UO2 will have the same power pro-file for the same amount of reactivity being inserted is dubious,but without performing neutronic pin cell calculations to quan-tify the difference in rod worth, it a fair compromise. However,in order to characterize fuel concept behaviors under similaroperational transients, the oxide needs a higher temperature.The power profile is scaled to a larger maximum value with thegoal of obtaining a maximum cladding temperature consistentwith the silicide case.

The fuel centerline temperature at its largest value for eachcase can be seen in Figure 4. The temperatures are normalizedto each fuel’s respective melting temperature (1938 K forsilicide and 3138 K in oxide). Silicide is consistently closer tofailure when even compared to the higher power profile oxidecase, though this maximum is reached much more quickly.Both temperature cases demonstrate that the silicide reacheshigher temperatures than the oxide in accident conditions, butit reaches its peak temperature more quickly. The increasedheat transfer rate of the silicide could mean the coolant channelspends less time in two phase flow conditions; arguably a moreimportant factor to consider in fuel element failure assessmentsfor RIAs.

A comparison of vapor volume fraction between the sili-cide and oxide fuel systems at the coolant channel outlet ismade in Figure 5. The silicide system boils more quicklythan the oxide and at around 250 seconds begins to reduceits volume fraction of vapor more quickly than the oxide. Atthe same power profile, the silicide takes slightly longer torecover to steady-state single phase flow according to thissimulations, but an important factor to keep in mind is thatthe volume fractions are highly dependent upon the wall heattransfer coefficient between the cladding and coolant. At themoment, single value convective wall heat transfer coefficientsare iteratively determined for each case. Only a single Hwvalue is utilized axially along the fuel element; so these resultsshould be treated as preliminary but do demonstrate generalbehaviors of the fuel concepts close to when the reactivityinsertion accident occurs.

2. Sensitivity Analysis

After perturbing parameters, the largest maximumcladding outer surface temperature obtained is 1146 K andsmallest is 1121 K. Results from the oxide case sensitivityanalysis are presented in Table II. As a comparison of simi-lar temperature profiles on the clad surface is desired, theseand any additional oxide results are generated from the higherpower profile oxide case. The least squares fit gives the highestR2 value, but cross-validated elastic net and cross-validatedlasso methods show very similar results. Several unimportantvariables have zero coefficients as the L1 penalty has selected asubset of more sensitive parameters. Also note that the highestL1 ratio is chosen by the cross validation scheme applied tothe elastic net model, giving results as close to the lasso as that

M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

Least Squares CV Elastic Net CV LassoR2 0.99750 0.99748 0.99747L1 Ratio - 0.95 (1)α - 3.74359 × 10−4 8.28625 × 10−4

pliq.,init. -1.32187e+00 -1.31440e+00 -1.31346e+00vliq.,init. 1.87986e-04 0.00000e+00 -0.00000e+00Tliq.,init. 2.08947e-02 1.79273e-02 9.83901e-03ρ f uel -4.96192e-01 -4.91427e-01 -4.86560e-01dgap width 2.40889e+00 2.41439e+00 2.41166e+00dclad thick. -2.71954e+00 -2.69626e+00 -2.69126e+00pplen.,init. 1.21873e-01 1.15416e-01 1.09265e-01Rclad -1.60707e-01 -1.55657e-01 -1.48612e-01Rfuel -4.65529e-01 -4.59022e-01 -4.52863e-01fk,fuel 2.83912e+00 2.82547e+00 2.82650e+00fcp,fuel -5.14804e+00 -5.12957e+00 -5.13677e+00fk,clad 6.04572e-03 7.19987e-04 0.00000e+00

TABLE II. Linear regression model coefficients for high R2

models for UO2 normalized using standard deviation (mostsensitive parameters are highlighted).

model is allowed. The most sensitive parameters determinedby these models are clad thickness, gap width, and scalingfactors for specific heat and the heat transfer coefficient.

The maximum surface cladding temperature for the per-turbed silicide cases is 1146 K and the minimum is 1117 K;so the silicide has a larger range for this output parametercompared to the oxide. Model characteristics utilized for thesensitivity analysis of the silicide case are found in Table III.The R2 values for these models are less confident about theirpredictive capability, but do provide insight into clad temper-ature sensitivities. Once again, the unpenalized linear leastsquares model performs better than the shrinkage methods.The three the most sensitive variables (gap width, claddingthickness, and scaling factor of fuel’s specific heat) are sharedbetween the silicide and oxide cases, and while the heat trans-fer coefficient scaling factor is considered a variable of lowimportance, its strong importance in the oxide case and sameorder of magnitude with the next most important variable afterthe three previously discussed allows for its use in the limitsurface search.

The extents and sensitivities of the fuel centerline tem-perature to the perturbed variables was also investigated. Theoxide case has maximum and minimum fuel centerline tem-peratures of 2037 and 1873 respectively. The parameters thecenterline temperature is most sensitive to in this case are fuelroughness, fuel k scaling factor, and fuel cp scaling factor.Perturbations in the maximum fuel centerline temperature ofsilicide case are more mild with a maximum of 1281 and min-imum of 1232. The important parameters of the silicide fueltemperature are fuel roughness, scaling factor of the fuel’sspecific heat, and the gap width.

3. Limit Surface Search

The limit surface search procedure is a time intensive pro-cess. The two fuel system simulations did not converge in time

Least Squares CV Elastic Net CV LassoR2 0.99202 0.99199 0.99194L1 Ratio - 0.95 (1)α - 4.89246 × 10−4 1.43564 × 10−3

pliq.,init. -9.45623e-01 -9.32882e-01 -9.22881e-01vliq.,init. -6.41105e-02 -6.12770e-02 -4.04804e-02Tliq.,init. 1.34611e-01 1.28800e-01 1.15525e-01ρ f uel -6.61286e-01 -6.51132e-01 -6.38406e-01dgap width 4.35585e+00 4.34161e+00 4.36625e+00dclad thick. -2.88561e+00 -2.87064e+00 -2.82809e+00pplen.,init. 1.61738e-03 -0.00000e+00 0.00000e+00Rclad 5.84397e-02 4.76140e-02 2.79713e-02Rfuel 2.42991e-01 2.30424e-01 2.15029e-01fk,fuel 2.05436e-01 1.99115e-01 1.84492e-01fcp,fuel -6.87210e+00 -6.83925e+00 -6.85280e+00fk,clad 1.43223e-02 1.57040e-02 8.07975e-04

TABLE III. Linear regression model coefficients for high R2

models for U3Si2 normalized using standard deviation (mostsensitive parameters are highlighted).

for results to be presented here, but limit surface identificationis well on its way to being completed. However, preliminaryresults generated as the search is in progress can be analyzed.Figure 6 shows the limit surface identified so far with 1140 Kas the failure criteria. While there are 663 data points in thevisualization, only 458 independent coupled simulations havebeen run as the SVM reduced order model generates predic-tions for the surface location. The three spatial dimensionsrepresent gap width, the fuel thermal conductivity scaling fac-tor, and the fuel specific heat scaling factor. The last of the fourinput variables perturbed is visualized as colormapped datapoints for the clad thickness. One can see that as this last vari-able increases in value, the failure surface moves negativelyin the gap width and fuel cp scaling factor spatial dimensionsand positively compared to fuel k scaling factor.

Another method of visualizing high dimensional datais demonstrated in Figure 7. Input variable coordinates arenormalized on the plot axes and each point identified as lyingon either the failure or pass limit surface is plotted as a line.The line color is either blue for passing points or red forfailure. The trends identified in the sensitivity analysis areverified here: the fuel specific heat scaling factor has a stronglynegative correlation to failure (higher temperatures) while gapwidth and k scale factor somewhat less strongly correspondto positively to temperature and clad thickness has a negativecorrelation.

The currently identified limit surface for clad surface max-imum temperature of the uranium silicide concept is shownin Figure 8. This surface is generated from the SVM reducedorder model utilizing 536 uranium silicide RIA simulationresults. This surface has many points sampled on the higherend of the clad thickness input variable’s sampling distribution,and appears to have a less regularly predicted structure than theoxide’s limit surface. Figure 9 is the parallel coordinates plotof the failure and pass limit surfaces. Again, the predicted sen-sitivites are evidenced from the failure points’ correlation with

M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

Fig. 6. Visualization of UO2 failure limit surface.

Fig. 7. Parallel coordinates of UO2 sensitive parameters alonglimit surface.

the positive or negative sensitivity and sensitivity magnitude.

IV. CONCLUSIONS

By quantifying CAFs using advanced simulation capabili-ties for use in an existing ATF evaulation framework, potentialfuel concepts may be appraised using quantitative metrics.The quantification methodology is not meant to be tied to onetool or analysis method, but as a demonstration of the processto quantify CAFs, the thermal behavior of silicide and oxidefuels are compared during a RIA simulated using coupled BI-SON and RELAP-7. The results presented illustrate that thesecodes are capable of running in RIA conditions as well as thatsensitivity investigations and limit surface quantification areviable avenues to quantify failure probabilities. In the courseof obtaining failure probabilities several vulnerabilities of thesilicide concept have been identified including higher fuel andcladding temperatures as well as a lack of information criticalto assessing the case (e.g. rod worths of silicide compared tothe oxide in RIA conditions, accurate long term two-phaseflow results). Converged limit surface run results are forth-

Fig. 8. Visualization of U3Si2 failure limit surface.

Fig. 9. Parallel coordinates of U3Si2 sensitive parametersalong limit surface.

coming and a prediction for failure probability can then bemade allowing the concept assessment factor to be quantified.

V. ACKNOWLEDGMENTS

The phrase, "No man is an island," has never been asevident as in the case of this work. At many points in thisproject, people offered advice, helped overcome challenges,or suggested a path forward to advance the work. Sincerestthanks go out to everyone named (and unnamed) whose inputshaped what was accomplished. Charlie Folsom and RichWilliamson provided excellent resources for getting started inmodeling RIAs in BISON, and it was thanks to Al Casagrandaalone that patch size issues encountered were diagnosed. Ad-ditional BISON troubleshooting and convergence issues weretackled with the aid of Steve Novascone. Without David An-drs, RELAP-7 would not exist as it currently does, and hiseven-tempered explanations were of great help to the thermo-hydraulically challenged. RAVEN developers continue to be ahelpful, enthusiastic team of data minded individuals. Specifi-cally, Cristian Rabiti suggested utilizing RAVEN to computelimit surfaces and calculate failure probailities and Andrea

M&C 2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea, April 16-20, 2017, on USB (2017)

Alfonsi made these analyses possible through the vast amountof knowledge (not to mention number of merge requests) hehas committed to RAVEN. Too many additional smaller con-tributions were offered to list individually, but thanks go out tothe MOOSE team, summer interns, and officemates for all oftheir expertise, encouragement, and fellowship. The greatestappreciation goes to Jason Hales and Rich Martineau for theirmentorship and the opportunity to be part of the great thingsgoing on at INL.

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